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Over a fixed commutative base ring, a compatible unital associative algebra and counital coassociative coalgebra.
Compatibility may equivalently be stated by requiring multiplication and unit to be coalgebra maps, or comultiplication and counit to be algebra maps.
wp-bialgebranlab-bialgebrastacks-algebra-textEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require comultiplication and counit to respect multiplication and unit.
Feed several structures into a construction junction and impose compatibility between them.
wp-bialgebranlab-bialgebrastacks-algebra-textEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Select bialgebras for which a convolution inverse of the identity exists; the antipode is then unique.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-hopf-algebranlab-hopf-algebrastacks-algebra-text